In Eq. (REF ), the penetrability factor depends on the channel radius \(R\) , orbit angular momentum \(\ell \)  and incident center-of-mass energy \(E\) .
In the treatment of King et al., a channel radius of \(R\) =4.1 fm [i.e., \(R\) =\(r_0\) (A\(_1^{1/3}\) +A\(_2^{1/3}\) ) with \(r_0\) =1.41 fm] was utilized in both
\(n\) +\(^7\) Be and \(p\) +\(^7\) Li systems. In the present calculation, the penetrability factor \(P_\ell (E,R)\)  for \(p\) +\(^7\) Li is calculated by an RCWFN
code [1]}, and that for neutron is calculated using Eq. (REF ) with \(F_\ell \)  and \(G_\ell \)  values tubulated by Feshbach and Lax [2]}.
Since the \(\alpha \)  particle is the spinless boson, the wavefunction for two identical \(\alpha \)  particles must be symmetric under interchange. However, the
wavefunction for an \(\ell \) =odd state in \(^8\) Be is antisymmetric by interchanging the two \(\alpha \)  particles. This implies that the compound state in
\(^8\) Be must have even parity for the incident \(\alpha \) +\(\alpha \)  channel. We know that both ground states for \(^7\) Li and \(^7\) Be have odd parity, and hence the
relative orbit angular momentum \(\ell \)  must be odd. Since the orbit centrifugal barrier [\(\propto \)\(\ell \) (\(\ell \) +1)] increases with respect to \(\ell \) ,
the \(p\) -wave (\(\ell \) =1) capture will dominate both the \(n\) +\(^7\) Be and \(p\) +\(^7\) Li exit channels. For each energy points listed in Table REF , we have
calculated the \(^4\) He(\(\alpha \) ,\(n\) )\(^7\) Be (g.s.) reaction cross section, and the associated uncertainty is estimated by considering the uncertainties in both
\(r_0\)  (in range of 1.1–1.5 fm [3]}) and incident energy (\(\pm \) 100 keV [4]}, [5]}). Finally, with the detailed-balance principle [6]}
the cross section of \(^7\) Be(\(n\) ,\(\alpha \) )\(^4\) He has been derived as listed in the third and fourth columns of Table REF . Here, three data points are
listed for \(E_\mathrm {c.m.}\)\(<\) 0.1 MeV, which are derived based on the experimental data [7]} of \(^7\) Li(\(p\) ,\(\alpha \) )\(^4\) He simply by the first
term in Eq. REF . The associated uncertainties are estimated by taking those of \(r_0\)  and incident energies into account. Alternatively, the low-energy
data of \(^4\) He(\(\alpha \) ,\(n\) )\(^7\) Be have been simply estimated by linearly interpolating two data points of \(^4\) He(\(\alpha \) ,\(p\) )\(^7\) Li at 37.48 MeV [5]}
and 38.09 MeV [9]}, and ultimately converted to those of \(^7\) Be(\(n\) ,\(\alpha \) )\(^4\) He. And we find that the interpolated results agree well with these three
data points at \(E_\mathrm {c.m.}\)\(<\) 0.1 MeV within uncertainties.
